We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let $n\le m$ and let $M$ be an oriented Riemannian $n$-manifold, $N$ a Riemannian $m$-manifold, and $\omega \in \Omega^n(N)$ a smooth closed non-vanishing $n$-form on $N$. A continuous Sobolev map $f\colon M \to N$ in $W^{1,n}_{\mathrm{loc}}(M,N)$ is a $K$-quasiregular $\omega$-curve for $K\ge 1$ if $f$ satisfies the distortion inequality $(\lVert\omega\rVert\circ f)\lVert Df\rVert^n \le K (\star f^* \omega)$ almost everywhere in $M$. We prove that quasiregular curves satisfy Gromov's quasiminimality condition and a version of Liouville's theorem stating that bounded quasiregular curves $\mathbb R^n \to \mathbb R^m$ are constant. We also prove a limit theorem that a locally uniform limit $f\colon M \to N$ of $K$-quasiregular $\omega$-curves $(f_j \colon M\to N)$ is also a $K$-quasiregular $\omega$-curve. We also show that a non-constant quasiregular $\omega$-curve $f\colon M \to N$ is discrete and satisfies $\star f^*\omega >0$ almost everywhere, if one of the following additional conditions hold: the form $\omega$ is simple or the map $f$ is $C^1$-smooth.

An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group.

In 1985, Robert L. Pego characterized compact families in $L^2(\reals)$ in terms of the Fourier transform. It took nearly 30 years to realize that Pego's result can be proved in a wider setting of locally compact abelian groups (works of Gorka and Kostrzewa). In the current paper, we argue that the Fourier transform is not the only integral transform that is efficient in characterizing compact families and suggest the Laplace transform as a possible alternative.

In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.